When mounted on the skin, modern sensors, circuits, radios, and power supply systems have the potential to provide clinical-quality health monitoring capabilities for continuous use, beyond the confines of traditional hospital or laboratory facilities. The most well-developed component technologies are, however, broadly available only in hard, planar formats. As a result, existing options in system design are unable to effectively accommodate integration with the soft, textured, curvilinear, and time-dynamic surfaces of the skin. Here, we describe experimental and theoretical approaches for using ideas in soft microfluidics, structured adhesive surfaces, and controlled mechanical buckling to achieve ultralow modulus, highly stretchable systems that incorporate assemblies of high-modulus, rigid, state-of-the-art functional elements. The outcome is a thin, conformable device technology that can softly laminate onto the surface of the skin to enable advanced, multifunctional operation for physiological monitoring in a wireless mode.